# To what extent is pi non-repeating? [closed]

I've been told that pi is an irrational (infinite and non-repeating) number.

But to what extent is it non-repeating?

It obviously repeats individual numbers, and I find it hard to believe that it doesn't repeat 2-3 digit sections eventually.

• A repeating decimal has a sequence of digits that is repeated ad infinitum after a certain point. $\pi$ has none such. This doesn't mean that it can't have subsequences repeating at different places, which it of course has.
– dxiv
Mar 17 '17 at 4:28
• Wow, three and a half hours and no answer. You must be angry.
– zhw.
Mar 17 '17 at 4:36

I've been told that pi is an irrational ,(infinite and non repeating), number. But to what extent is it non repeating. It obviously repeats individual numbers, and i find it hard to believe that it doesn't repeat 2-3 digit sections eventually.

$\pi$ certainly does repeat 2-3 digit sections eventually. There are only 1,000 different sequences of 3 digits, so there's no way that $\pi$ (or any other number) can avoid repeating some of them.

In fact, if $\pi$ is a so-called normal number, then every possible 3-digit sequence appears infinitely many times (as does every 10-digit sequence, every 1,000,000-digit sequence, and so on).

When we say that the decimal expansion of $\pi$ is "non-repeating", what we mean is that $\pi$ never begins to repeat just one sequence of digits over and over forever. In other words, the decimal expansion of $\pi$ can repeat itself; it just can't ever fall into a cycle where it repeats the same thing forever.

A number is said to be repeating if its digit sequence eventually gets periodic, that is, it repeats the same subsequence infinitely often without anything in between.

For example, the following is a repeating number: $$1.269\overline{13} = 1.269\color{red}{1313131313131313131313}\ldots$$ The repeating digit string here is, of course, $13$.

However the following number is not repeating, despite containing $13$ infinitely often: $$1.269\color{red}{13}1\color{red}{13}2\color{red}{13}4\color{red}{13}8\color{red}{13}16\color{red}{13}32\color{red}{13}64\color{red}{13}128\color{red}{13}256\color{red}{13}512\color{red}{13}1024\color{red}{13}\ldots$$

Also note that the question to what extent a number is not repeating doesn't actually make sense: Either it is repeating, or it is not.

Non-repeating in this sense just means that it doesn't repeat forever. For instance, pi could repeat the sequence "284" five hundred times in a row and still be irrational, as long as it eventually does not repeat.

Another property that might help you understand is the property that every number sequence is contained in pi. Numbers which satisfy this property are called normal numbers, and pi is suspected to be but not proved to be normal. This stackexchange post goes into more detail.

• This is not correct - normality is a much stronger property than just having every sequence occur. Mar 17 '17 at 7:31